# Documentation

Lean.Meta.ExprDefEq

Support for Lean.reduceBool and Lean.reduceNat

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Support for reducing Nat basic operations.

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Support for constraints of the form ("..." =?= String.mk cs)

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Return true if e is of the form fun (x_1 ... x_n) => ?m x_1 ... x_n), and ?m is unassigned. Remark: n may be 0.

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• Failed to establish that explicit arguments are def-eq. Remark: higher output parameters, and parameters that depend on them are postponed.

failed: Lean.Meta.DefEqArgsFirstPassResult
• Succeeded. The array postponedImplicit contains the position of the implicit arguments for which def-eq has been postponed. postponedHO contains the higher order output parameters, and parameters that depend on them. They should be processed after the implict ones. postponedHO is used to handle applications involving functions that contain higher order output parameters. Example:

getElem :
{cont : Type u_1} → {idx : Type u_2} → {elem : Type u_3} →
{dom : cont → idx → Prop} → [self : GetElem cont idx elem dom] →
(xs : cont) → (i : idx) → (h : dom xs i) → elem
→ {idx : Type u_2} → {elem : Type u_3} →
{dom : cont → idx → Prop} → [self : GetElem cont idx elem dom] →
(xs : cont) → (i : idx) → (h : dom xs i) → elem
→ {elem : Type u_3} →
{dom : cont → idx → Prop} → [self : GetElem cont idx elem dom] →
(xs : cont) → (i : idx) → (h : dom xs i) → elem
→
{dom : cont → idx → Prop} → [self : GetElem cont idx elem dom] →
(xs : cont) → (i : idx) → (h : dom xs i) → elem
→ idx → Prop} → [self : GetElem cont idx elem dom] →
(xs : cont) → (i : idx) → (h : dom xs i) → elem
→ Prop} → [self : GetElem cont idx elem dom] →
(xs : cont) → (i : idx) → (h : dom xs i) → elem
→ [self : GetElem cont idx elem dom] →
(xs : cont) → (i : idx) → (h : dom xs i) → elem
→
(xs : cont) → (i : idx) → (h : dom xs i) → elem
→ (i : idx) → (h : dom xs i) → elem
→ (h : dom xs i) → elem
→ elem


The argumengs dom and h must be processed after all implicit arguments otherwise higher-order unification problems are generated. See issue #1299, when trying to solve

getElem ?a ?i ?h =?= getElem a i (Fin.val_lt_of_le i ...)


we have to solve the constraint

?dom a i.val =?= LT.lt i.val (Array.size a)


by solving after the instance has been synthesized, we reduce this constraint to a simple check.

ok:

Result type for isDefEqArgsFirstPass.

Instances For
@[specialize #[]]
def Lean.Meta.isDefEqBindingDomain (fvars : ) (ds₂ : ) (k : ) :

Check whether the types of the free variables at fvars are definitionally equal to the types at ds₂.

Pre: fvars.size == ds₂.size

This method also updates the set of local instances, and invokes the continuation k with the updated set.

We can't use withNewLocalInstances because the isDeq fvarType d₂ may use local instances.

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partial def Lean.Meta.isDefEqBindingDomain.loop (fvars : ) (ds₂ : ) (k : ) (i : Nat) :

Each metavariable is declared in a particular local context. We use the notation C |- ?m : t to denote a metavariable ?m that was declared at the local context C with type t (see MetavarDecl). We also use ?m@C as a shorthand for C |- ?m : t where t is the type of ?m.

The following method process the unification constraint

   ?m@C a₁ ... aₙ =?= t


We say the unification constraint is a pattern IFF

1) a₁ ... aₙ are pairwise distinct free variables that are ​*not*​ let-variables.
2) a₁ ... aₙ are not in C
3) t only contains free variables in C and/or {a₁, ..., aₙ}
4) For every metavariable ?m'@C' occurring in t, C' is a subprefix of C
5) ?m does not occur in t
​*not*​ let-variables.
2) a₁ ... aₙ are not in C
3) t only contains free variables in C and/or {a₁, ..., aₙ}
4) For every metavariable ?m'@C' occurring in t, C' is a subprefix of C
5) ?m does not occur in t
​ let-variables.
2) a₁ ... aₙ are not in C
3) t only contains free variables in C and/or {a₁, ..., aₙ}
4) For every metavariable ?m'@C' occurring in t, C' is a subprefix of C
5) ?m does not occur in t


Claim: we don't have to check free variable declarations. That is, if t contains a reference to x : A := v, we don't need to check v. Reason: The reference to x is a free variable, and it must be in C (by 1 and 3). If x is in C, then any metavariable occurring in v must have been defined in a strict subprefix of C. So, condition 4 and 5 are satisfied.

If the conditions above have been satisfied, then the solution for the unification constrain is

?m := fun a₁ ... aₙ => t


Now, we consider some workarounds/approximations.

A1) Suppose t contains a reference to x : A := v and x is not in C (failed condition 3) (precise) solution: unfold x in t.

A2) Suppose some aᵢ is in C (failed condition 2) (approximated) solution (when config.quasiPatternApprox is set to true) : ignore condition and also use

    ?m := fun a₁ ... aₙ => t


Here is an example where this approximation fails: Given C containing a : nat, consider the following two constraints ?m@C a =?= a ?m@C b =?= a

If we use the approximation in the first constraint, we get ?m := fun x => x when we apply this solution to the second one we get a failure.

IMPORTANT: When applying this approximation we need to make sure the abstracted term fun a₁ ... aₙ => t is type correct. The check can only be skipped in the pattern case described above. Consider the following example. Given the local context

  (α : Type) (a : α)


we try to solve

 ?m α =?= @id α a


If we use the approximation above we obtain:

 ?m := (fun α' => @id α' a)


which is a type incorrect term. a has type α but it is expected to have type α'.

The problem occurs because the right hand side contains a free variable a that depends on the free variable α being abstracted. Note that this dependency cannot occur in patterns.

We can address this by type checking the term after abstraction. This is not a significant performance bottleneck because this case doesn't happen very often in practice (262 times when compiling stdlib on Jan 2018). The second example is trickier, but it also occurs less frequently (8 times when compiling stdlib on Jan 2018, and all occurrences were at Init/Control when we define monads and auxiliary combinators for them). We considered three options for the addressing the issue on the second example:

A3) a₁ ... aₙ are not pairwise distinct (failed condition 1). In Lean3, we would try to approximate this case using an approach similar to A2. However, this approximation complicates the code, and is never used in the Lean3 stdlib and mathlib.

A4) t contains a metavariable ?m'@C' where C' is not a subprefix of C. If ?m' is assigned, we substitute. If not, we create an auxiliary metavariable with a smaller scope. Actually, we let elimMVarDeps at MetavarContext.lean to perform this step.

A5) If some aᵢ is not a free variable, then we use first-order unification (if config.foApprox is set to true)

   ?m a_1 ... a_i a_{i+1} ... a_{i+k} =?= f b_1 ... b_k


reduces to

   ?M a_1 ... a_i =?= f
a_{i+1}        =?= b_1
...
a_{i+k}        =?= b_k


A6) If (m =?= v) is of the form

    ?m a_1 ... a_n =?= ?m b_1 ... b_k

then we use first-order unification (if config.foApprox is set to true)


A7) When foApprox, we may use another approximation (constApprox) for solving constraints of the form ?m s₁ ... sₙ =?= t where s₁ ... sₙ are arbitrary terms. We solve them by assigning the constant function to ?m. ?m := fun _ ... _ => t

 In general, this approximation may produce bad solutions, and may prevent coercions from being tried.
For example, consider the term pure (x > 0) with inferred type ?m Prop and expected type IO Bool.
In this situation, the
elaborator generates the unification constraint

?m Prop =?= IO Bool

It is not a higher-order pattern, nor first-order approximation is applicable. However, constant approximation
produces the bogus solution ?m := fun _ => IO Bool, and prevents the system from using the coercion from
the decidable proposition x > 0 to Bool.

On the other hand, the constant approximation is desirable for elaborating the term

let f (x : _) := pure "hello"; f ()

with expected type IO String.
In this example, the following unification contraint is generated.

?m () String =?= IO String

It is not a higher-order pattern, first-order approximation reduces it to

?m () =?= IO

which fails to be solved. However, constant approximation solves it by assigning

?m := fun _ => IO

Note that fs type is (x : ?α) -> ?m x String. The metavariable ?m may depend on x.
If constApprox is set to true, we use constant approximation. Otherwise, we use a heuristic to decide
whether we should apply it or not. The heuristic is based on observing where the constraints above come from.
In the first example, the constraint ?m Prop =?= IO Bool come from polymorphic method where ?m is expected to
be a **function** of type Type -> Type. In the second example, the first argument of ?m is used to model
a **potential** dependency on x. By using constant approximation here, we are just saying the type of f
does **not** depend on x. We claim this is a reasonable approximation in practice. Moreover, it is expected
by any functional programmer used to non-dependently type languages (e.g., Haskell).
We distinguish the two cases above by using the field numScopeArgs at MetavarDecl. This fiels tracks
how many metavariable arguments are representing dependencies.

def Lean.Meta.mkAuxMVar (lctx : Lean.LocalContext) (localInsts : Lean.LocalInstances) (type : Lean.Expr) (numScopeArgs : ) :
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@[inline]
def Lean.Meta.CheckAssignment.run (mvarId : Lean.MVarId) (fvars : ) (hasCtxLocals : Bool) (v : Lean.Expr) :
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partial def Lean.Meta.CheckAssignment.assignToConstFun (mvar : Lean.Expr) (numArgs : Nat) (newMVar : Lean.Expr) :

Auxiliary function used to "fix" subterms of the form ?m x_1 ... x_n where x_is are free variables, and one of them is out-of-scope. See Expr.app case at check. If ctxApprox is true, then we solve this case by creating a fresh metavariable ?n with the correct scope, an assigning ?m := fun _ ... _ => ?n

partial def Lean.Meta.CheckAssignment.checkAssignmentAux (mvarId : Lean.MVarId) (fvars : ) (hasCtxLocals : Bool) (v : Lean.Expr) :
def Lean.Meta.CheckAssignmentQuick.check (hasCtxLocals : Bool) (mctx : Lean.MetavarContext) (lctx : Lean.LocalContext) (mvarDecl : Lean.MetavarDecl) (mvarId : Lean.MVarId) (fvars : ) (e : Lean.Expr) :
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partial def Lean.Meta.CheckAssignmentQuick.check.visit (hasCtxLocals : Bool) (mctx : Lean.MetavarContext) (lctx : Lean.LocalContext) (mvarDecl : Lean.MetavarDecl) (mvarId : Lean.MVarId) (fvars : ) (e : Lean.Expr) :
def Lean.Meta.checkAssignment (mvarId : Lean.MVarId) (fvars : ) (v : Lean.Expr) :

Auxiliary function for handling constraints of the form ?m a₁ ... aₙ =?= v. It will check whether we can perform the assignment

?m := fun fvars => v


The result is none if the assignment can't be performed. The result is some newV where newV is a possibly updated v. This method may need to unfold let-declarations.

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@[inline]
def Lean.Meta.whenUndefDo (x : ) (k : ) :
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@[export lean_is_expr_def_eq]